Abstract : The thesis is concerned with proving the the eigenvalues of a specific unsymmetric matrix are real and positive, without actually computing them. The method of finite differences is applied to the vibration analysis of a cantilever beam and leads to an unsymmetric stiffness matrix in the eigenvalue problem formulation. The technique employed in the proof is based on perturbation theory given by Wilkinson for real symmetric matrices. Application of the theory is made to the cantilever beam eigenvalue problem. The results verify that the eigenvalues of this and other unsymmetric matrices can be proven real and positive without their actual values being calculated. (Author)